Enhanced lattice reduction systems and methods

ABSTRACT

An exemplary embodiment of the present invention provides a lattice reduction method comprising obtaining a preliminary estimate of a transformation matrix, generating a covariance matrix based on the preliminary estimate of the transformation matrix, reducing diagonal elements of the covariance matrix to generate a unimodular transformation matrix, and using the unimodular transformation matrix to obtain an estimate of an input.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 61/453,623, filed on 17 Mar. 2011, which is incorporated herein byreference in its entirety as if fully set forth below.

TECHNICAL FIELD OF THE INVENTION

Embodiments of the present invention relate generally to signalprocessing systems and methods and, more particularly, to systems,devices, and methods for lattice reduction.

BACKGROUND OF THE INVENTION

Multiple-Input Multiple-Output (“MIMO”) communication systems arebecoming increasingly popular as a solution to increasing demands forhigher data-rates and more reliable wireless communication systems.These systems comprise multiple antennas at a transmitter side of thecommunication system and multiple antennas at the receiver side of thecommunication system. Each transmitter antenna can transmit a differentsignal at a common frequency through a different channel of thecommunication system. Each receiver antenna may receive each signal fromthe multiple transmitter-antennas. During transit, the transmittedsignals may encounter different obstacles such that the frequencyresponse of each channel is different. The plurality of channels used inthe transmission of symbols from the plurality of transmitter antennasto the plurality of receiver antennas together form a channel matrix.The input-output relationship of a typical MIMO system can berepresented by Equation 1.

y=Hs+w   Equation 1:

In Equation 1, y represents an M×1 output vector received by thereceiver antennas, s represents an N×1 input symbol vector transmittedby the transmitter antennas, H represents an M×N channel matrix, and wrepresents an unknown noise vector.

A common goal of conventional systems is to attempt to efficientlydetect the transmitted symbol vector s by determining frequency responseof each channel in the communication system, i.e. accurately estimatingthe channel matrix H.

It can be assumed that the elements of noise vector w are independentlydistributed with each entry of the noise vector being a random variablewith zero mean and variance σ_(ω) ². Given the known distribution of thenoise vector, the optimal solution to the MIMO symbol detection problemis Maximum Likelihood (“ML”) detection. ML detection, however, requiresan exhaustive search over all possible transmitted symbol vectors,requiring high computational complexity. This approach is infeasible forhardware implementations when either a large signal constellation or alarge number of transmit and receive antennas are employed. Hence, agoal of conventional systems is to design hardware for MIMO symboldetection that achieves comparable Bit-Error-Rate (“BER”) performance tothe ML detector while having low hardware complexity and meetingthroughput and latency requirements, especially as the size of the MIMOsystem increases.

Some conventional MIMO symbol detections systems employ methods oflinear detection and Successive Interference Cancelation (“SIC”).Because most of the required processing for these detectors need onlyoccur at the maximum packet-rate (preprocessing) and the requiredsymbol-rate processing has relatively low-complexity, the throughputrequirements for certain wireless standards, such as 802.11n, can beachieved in these systems. These conventional systems, however, do notcollect the same diversity (negative logarithmic asymptotic slope of theBER versus Signal-to-Noise-Ratio (“SNR”) curve) as ML detection. As aresult, these methods exhibit greatly reduced system performancecompared to ML detectors

Other conventional symbol detection systems employ Sphere Decoding(“SD”) algorithms. Hardware implementations of SD algorithms can achieveML or near-ML performance. Unfortunately, these methods exhibit greatlyincreased symbol-rate processing complexity compared to linear or SICdetectors. The complexity of SD methods can also vary widely withchanging channel conditions.

The maximum packet-rate of 802.11n is considerably less than thesymbol-rate. Therefore, it is desirable to obtain detection systems andmethods that achieve ML or near-ML performance at the cost of increasedpreprocessing complexity as opposed to increased symbol-rate processingcomplexity. Systems having these desired characteristics include LatticeReduction (“LR”) aided detectors, which, unlike SD methods, incorporateLR algorithms into the preprocessing part of linear or SIC detectors andincrease the symbol-rate processing complexity slightly. Specifically,LR systems and methods employ lattice reduction once per received packet(per subcarrier). LR-aided detectors also exhibit the desirable propertyof having a complexity that is independent of both the channel SNR andsignal constellation (assuming individual arithmetic operations haveO(1) complexity).

A variety of hardware realizations of LR-aided detectors have beenexplored to exploit these properties and to achieve near-ML performance.Various explorations have included a VLSI implementation of a simplifiedBrun's LR algorithm and a software implementation of Seysen's LRalgorithm on a reconfigurable baseband processor. Other conventionalLR-aided detectors employ variations the Complex Lenstra-Lenstra-Lovász(“CLLL”) LR algorithm. Unfortunately, the performance of theseconventional LR-aided detectors decreases and their complexity increasesas the size of the MIMO system increases, i.e. the number of transmitterand receiver antennas increases.

Accordingly, there is a desire for more efficient and less complexLR-aided detection systems and methods. Various embodiments of thepresent invention address these desires.

BRIEF SUMMARY OF THE INVENTION

The present invention relates to enhanced LR systems and methods. Anexemplary embodiment of the present invention provides a LR methodcomprising obtaining a preliminary estimate of a transformation matrix,generating a covariance matrix based on the preliminary estimate of thetransformation matrix, reducing diagonal elements of the covariancematrix to generate a unimodular transformation matrix, and using theunimodular transformation matrix to obtain an estimate of an input. Inan exemplary embodiment of the present invention, the method furthercomprises obtaining a signal at a plurality of input terminals, thesignal representing the input linearly transformed by the transformationmatrix. In some exemplary embodiments of the present invention, themethod further comprises repeating the step of reducing the diagonalelements until the diagonal elements of the covariance matrix are nolonger reducible. In an exemplary embodiment of the present invention,the input comprises a symbol vector. In another exemplary embodiment ofthe present invention, the transformation matrix comprises a channelmatrix.

In an exemplary embodiment of the present invention, reducing diagonalelements of the covariance matrix comprises reducing a largest reduciblediagonal element of the covariance matrix. In another exemplaryembodiment of the present invention, reducing diagonal elements of thecovariance matrix comprises reducing a smallest reducible diagonalelement of the covariance matrix. In yet another exemplary embodiment ofthe present invention, reducing diagonal elements of the covariancematrix comprises reducing a random reducible diagonal element of thecovariance matrix. In still yet another exemplary embodiment of thepresent invention, reducing diagonal elements of the covariance matrixcomprises minimizing cost by reducing a reducible diagonal element ofthe covariance matrix requiring the least cost to be found. Further, insome exemplary embodiments of the present invention, reducing diagonalelements of the covariance matrix comprises an iterative process.

Another exemplary embodiment of the present invention provides a LRmethod comprising obtaining a preliminary estimate of a transformationmatrix, generating a covariance matrix based on the preliminary estimateof the transformation matrix, and reducing diagonal elements of thecovariance matrix to generate a unimodular transformation matrix. In yetanother exemplary embodiment of the present invention, obtaining apreliminary estimate of a transformation matrix comprises accessing datafrom a memory.

In addition to LR methods, the present invention is directed to LRsystems. An exemplary embodiment of the present invention provides a LRsystem comprising a plurality of input terminals, a processor, andlogic. In some exemplary embodiments of the present invention, theplurality of input terminals can be configured to obtain a signal, thesignal representing an input transformed by a transformation matrix. Insome exemplary embodiments of the present invention, the logic can bestored in a non-transitory computer readable media that can be executedby the processor. In an exemplary embodiment of the present invention,execution of the logic by the processor causes the system to obtain apreliminary estimate of the transformation matrix, generate a covariancematrix based on the preliminary estimate of the transformation matrix,reduce diagonal elements of the covariance matrix to generate aunimodular transformation matrix, and use the unimodular transformationmatrix to obtain an estimate of the input. In some exemplary embodimentsof the present invention, the logic is further configured to repeat thestep of reducing diagonal elements of the covariance matrix until thediagonal elements are no longer reducible. In an exemplary embodiment ofthe present invention, the input can comprise a symbol vector. Inanother exemplary embodiment of the present invention, thetransformation matrix comprises a channel matrix.

In an exemplary embodiment of the present invention, the logic can befurther configured to reduce diagonal elements of the covariance matrixby reducing a largest reducible diagonal element of the covariancematrix. In another exemplary embodiment of the present invention, thelogic can be further configured to reduce diagonal elements of thecovariance matrix by reducing a smallest reducible diagonal element ofthe covariance matrix. In yet another exemplary embodiment of thepresent invention, the logic can be further configured to reducediagonal elements of the covariance matrix by reducing a randomreducible diagonal element of the covariance matrix. In still yetanother exemplary embodiment of the present invention, the logic isfurther configured to minimize cost by reducing diagonal elements of thecovariance matrix by reducing a reducible diagonal element of thecovariance matrix requiring the least cost to be found. Further, in someexemplary embodiments of the present invention, the logic is furtherconfigured to reduce diagonal elements of the covariance matrix byiteratively reducing at least one diagonal element of the covariancematrix.

These and other aspects of the present invention are described in theDetailed Description of the Invention below and the accompanyingfigures. Other aspects and features of embodiments of the presentinvention will become apparent to those of ordinary skill in the artupon reviewing the following description of specific, exemplaryembodiments of the present invention in concert with the figures. Whilefeatures of the present invention may be discussed relative to certainembodiments and figures, all embodiments of the present invention caninclude one or more of the features discussed herein. While one or moreembodiments may be discussed as having certain advantageous features,one or more of such features may also be used with the variousembodiments of the invention discussed herein. In similar fashion, whileexemplary embodiments may be discussed below as system or methodembodiments, it is to be understood that such exemplary embodiments canbe implemented in various devices, systems, and methods of the presentinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The following Detailed Description of the Invention is better understoodwhen read in conjunction with the appended drawings. For the purposes ofillustration, there is shown in the drawings exemplary embodiments, butthe subject matter is not limited to the specific elements andinstrumentalities disclosed.

FIG. 1 provides block diagram of a lattice reduction method, inaccordance with an exemplary embodiment of the present invention.

FIG. 2 provides pseudo code for a lattice reduction method, inaccordance with an exemplary embodiment of the present invention.

FIGS. 3A-3C provide plots of BER performance of conventional methods andan exemplary embodiment of the present invention.

FIG. 4 provides plots of BER performance of conventional methods and anexemplary embodiment of the present invention.

FIG. 5 provides plots of BER performance of conventional methods and anexemplary embodiment of the present invention.

FIG. 6 provides plots of the average number of arithmetic operations forbasis updates in conventional methods and an exemplary embodiment of thepresent invention.

FIG. 7 provides plots of CCDF of od(H) and od({tilde over (H)}) for MIMOsystems with N=M=4 for conventional methods and an exemplary embodimentof the present invention.

FIG. 8 provides plots indicating performance comparisons of conventionalmethods and an exemplary embodiment of the present invention in MIMOsystems with 4 QAM, SNR=20 dB, and different numbers of antennas.

FIG. 9 provides plots indicating performance comparisons of conventionalmethods and an exemplary embodiment of the present invention in MIMOsystems with 64 QAM, SNR=30 dB, and different numbers of antennas.

FIG. 10 provides plots of the average number of basis updates forconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with different numbers of antennas.

FIG. 11 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with 256 QAM.

FIG. 12 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with 64 QAM, SNR=30 dB, and different numbersof antennas.

FIG. 13 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with M=N=64 and 256 QAM.

FIG. 14 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with M=N=32 and different constellation sizes.

FIG. 15 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with M=N=32, 64 QAM, and different SNRs.

DETAILED DESCRIPTION OF THE INVENTION

To facilitate an understanding of the principles and features of thepresent invention, various illustrative embodiments are explained below.In particular, the invention is described in the context of beinglattice reduction systems and methods. Embodiments of the presentinvention can be applied to many wireless MIMO communication systemstandards known in the art, including, but not limited to, IEEE 802.11(Wi-Fi), 4G, 3GPP, Long Tenn Evolution, Wi-MAX, HSPA+, and the like.Embodiments of the invention, however, are not limited to use inwireless MIMO communication systems. Rather, embodiments of theinvention can be used for processing other MIMO communication systems,including, but not limited to, optical MIMO systems or othertransmission systems having an architecture incorporating multipletransmitters and/or multiple transceivers. Embodiments of the presentinvention are not limited, however, to MIMO communication systems.Instead, various exemplary embodiments of the present invention can beapplied to many wireless communication systems, including, but notlimited to, SC-FDMA systems, GSC-FDMA systems, and the like.Additionally, various embodiments of the present invention can beapplied to many systems that may benefit from employing conventionallattice reduction systems and methods, including, but not limited to,GPS systems, cryptography systems, and the like.

The components described hereinafter as making up various elements ofthe invention are intended to be illustrative and not restrictive. Manysuitable components or steps that would perform the same or similarfunctions as the components or steps described herein are intended to beembraced within the scope of the invention. Such other components orsteps not described herein can include, but are not limited to, forexample, similar components or steps that are developed afterdevelopment of the invention.

A goal of LR aided detectors is to reduce the lattice basis of atransformation matrix, H, to find a “better” transformation matrix{tilde over (H)}=HT , where T is a unimodular matrix, such that entriesof T and T⁻¹ are Gaussian integers and the determinant of T is ±1 or ±j.To find the unimodular transformation matrix T and the “better”transformation matrix {tilde over (H)}, different methods have beenproposed, e.g. CLLL algorithm and Seysen's algorithm. After obtaining T,zero forcing (“ZF”) equalization can be performed with the “better”transformation matrix {tilde over (H)} as shown in Equation 2.

y={tilde over (H)}′x=T ⁻¹ s+{tilde over (H)}′η=z+n   Equation 2:

Because entries of s, T, and z are Gaussian integers, an estimate of zcan be obtained by quantizing the equalized signal as shown in Equation3.

{circumflex over (z)}=Q(y)   Equation 3:

An estimate of ŝ can then be obtained by rounding T{circumflex over (z)}with appropriate constellation as shown in Equation 4.

ŝ=Q(T{circumflex over (z)})   Equation 4:

Various embodiments of the present invention provide improved LR systemsand methods for determining a unimodular transformation matrix T and atransformation matrix {tilde over (H)}.

As shown in FIG. 1, an exemplary embodiment of the present inventionprovides an LR method 100. The method 100 comprises obtaining a signalat a plurality of input terminals 105. Unless expressly limited by itscontext, as used herein, the term “obtaining” indicates any of itsordinary meanings, such as sensing, measuring, generating, recording,receiving (e.g. from an antenna or another input terminal), accessing,or retrieving (e.g. from memory or another storage element). The inputterminals can be many input terminals known in the art, including, butnot limited to, receivers, antennas, optical inputs, data access points,pins on a chip or IC, registers or locations in memory, and the like.

The signal can represent an input s linearly transformed by atransformation matrix. As used herein, the transformation matrixmathematically represents at least a portion of the transformation of aninput due to interaction with one or more devices, conditions, or thelike. For example, in an exemplary embodiment, the input can comprise asymbol vector transmitted by a plurality of antennas and thetransformation matrix can comprise a channel matrix with datarepresentative of at least a portion of the transformations to one ormore portions of the symbol vector by wireless transmission through aplurality of channels (e.g. as represented in Equation 1). WhileEquation 1 references Gaussian distributed noise, various embodiments ofthe present invention also apply to non-Gaussian noise, which can beapproximated as a Gaussian distribution, e.g. by invoking the centrallimit theorem. In another exemplary embodiment of the present invention,the input can comprise an unencrypted data and the transformation matrixcan comprise data representative of at least a portion of thetransformations to the unencrypted data used in order to encrypt thedata.

The method 100 can further comprise obtaining a preliminary estimate ofthe transformation matrix H 110. In an exemplary embodiment of thepresent invention, the preliminary estimate of the transformation matrixH can be obtained using a channel estimator. The channel estimator canbe many channel estimators known in the art, including, but not limitedto, a least squares channel estimator. In some embodiments of thepresent invention, obtaining a preliminary estimate of thetransformation matrix 110 can comprise accessing data stored in memory.In some embodiments of the present invention, the data stored in memorycan be representative of the preliminary estimate of the transformationmatrix.

Instead of simply finding a more orthogonal channel matrix H as inconventional methods, some enhanced LR systems and methods of thepresent invention obtain H such that the asymptotic pairwise errorprobability (“PEP”) of a detector is reduced or minimized For example,consider the zero-forcing detector (“ZFD”) represented by Equation 5.

ŝ ^(ZF) =Q(H′x)=Q((H ^(H) H)⁻¹ H ^(H) x)   Equation 5

Given the ZFD in Equation 5, the ith transmitted symbol s_(i) can beerroneously detected as ŝ_(i)≠s_(i). The PEP given the transformationmatrix H is then represented by Equation 6.

$\begin{matrix}{{P\left( {s_{i}->\left. {\hat{s}}_{i} \middle| H \right.} \right)} = {Q\left( \sqrt{\frac{{e_{s_{i}}}^{2}}{2\sigma_{\omega}^{2}C_{i,i}}} \right)}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

In Equation 6, e_(s) _(i) =s_(i)−ŝ_(i) and

${Q(x)} = {\left( {2\pi} \right)^{- \frac{1}{2}}{\int_{x}^{\infty}{{\exp \left( {{- t^{2}}/2} \right)}{{t}.}}}}$

C=(H^(H)H)⁻¹ is a covariance matrix of the noise after equalization, andC_(i,i) can denote the (i,i)th element of C.

Similarly, the PEP for the LR-aided ZFD represented by Equation 2 giventhe “better” transformation matrix {tilde over (H)} is represented byEquation 7.

$\begin{matrix}{{P\left( {z_{i}->\left. {\hat{z}}_{i} \middle| \overset{\sim}{H} \right.} \right)} = {Q\left( \sqrt{\frac{{e_{z_{i}}}^{2}}{2\sigma_{\omega}^{2}{\overset{\sim}{C}}_{i,i}}} \right)}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

In Equation 7, e_(z) _(i) =z_(i)−{circumflex over (z)}_(i) and {tildeover (C)}=({tilde over (H)}^(H){tilde over (H)})^(−1=T) ⁻¹C(T⁻¹)^(H).Because the PEP for the ith symbol z_(i) is determined by {tilde over(C)}_(i,i), the PEP performance of z_(i) can be improved if the diagonalelements of {tilde over (C)} are reduced by using a lattice-basedmethod. Accordingly, in an exemplary embodiment of the presentinvention, the method further comprises generating a covariance matrix{tilde over (C)} 115 and reducing one or more diagonal elements of thecovariance matrix {tilde over (C)} 120. Additionally, in someembodiments of the present invention, reducing diagonal elements of thecovariance matrix {tilde over (C)} can generate/update a unimodulartransformation matrix T. The method can further comprise using theunimodular transformation matrix T to obtain an estimate of the input s125.

As the SNR approaches infinity, for all the PEPs corresponding to theith symbol, the one with the smallest

$\frac{{e_{z_{i}}}^{2}}{{\overset{\sim}{C}}_{i,i}}$

or similarly, the largest {tilde over (C)}_(i,i) , is the dominant termon PEP. Thus, some embodiments of the present invention seek to generatea unimodular matrix T and minimize the diagonal elements of thecovariance matrix {tilde over (C)}. Mathematically, this optimizationmethod can be represented by Equation 8.

$\begin{matrix}{\begin{matrix}\min \\{s.t.}\end{matrix}\begin{matrix}{\max \left( {\overset{\sim}{C}}_{i,i} \right)} \\{\overset{\sim}{C} = {T^{- 1}{C\left( T^{- 1} \right)}^{H}}}\end{matrix}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

In Equation 8, C=(H^(H)H)⁻¹ is the covariance matrix after ZFequalization (The present invention is also applicable to otherdetectors, e.g. an minimum mean square error detector (“MMSE-D”) inwhich C=(H^(H)H+σ²I)⁻¹). Solving the global optimal solution of Equation8 can be a non-deterministic polynomial hard (“NP-hard”) problem.Therefore, it can be impractical to solve Equation 8 globally.

Accordingly, exemplary embodiments of the present invention providemethods for finding a local optimal solution to Equation 8. In someembodiments of the present invention, one or more steps of in the method100 are iterative. For example, in each iteration, the method can reduceone of the diagonal elements of {tilde over (C)} by choosing an indexpair (i, k) and λ_(i,k)∈Z+Zj updates the unimodular matrix T′=(T⁻¹)^(H)with the column-addition operation as shown in Equation 9.

t′_(k)←t′_(k)+λ_(i,k)t′_(i)   Equation 9:

In Equation 9, t′_(k) is the kth column of T′. The corresponding updatesto the covariance matrix {tilde over (C)} can be represented by Equation10.

{tilde over (c)}_(k)←{tilde over (c)}_(k)+λ_(i,k){tilde over (c)}_(i)

{tilde over (c)}^((k))←{tilde over (c)}^((k))+λ*_(i,k){tilde over(c)}^((i))   Equation 10:

In Equation 10, {tilde over (c)}_(k) and {tilde over (c)}^((k)) are thekth column and kth row of {tilde over (C)}, respectively. In anexemplary embodiment of the present invention, for each iteration, themethod 100 can also choose λ_(i,k) as shown in Equation 11:

$\begin{matrix}{\lambda_{i,k} = {- \left\lbrack \frac{{\overset{\sim}{C}}_{i,k}}{{\overset{\sim}{C}}_{i,i}} \right\rbrack}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

In Equation 11, the brackets [ ] are used to indicate a roundingfunction that rounds the real and imaginary parts to closest integervalues. Thus, in an exemplary embodiment of the present invention, thereduced value for {tilde over (C)}_(k,k) is shown in Equation 12.

R _(i,k)=−|λ_(i,k)|² {tilde over (C)} _(i,i)−λ*_(i,k) {tilde over (C)}_(i,k)−λ_(i,k) {tilde over (C)} _(k,i)≧0

In some embodiments of the present invention, the method repeats theseiterations of reducing a diagonal element to generate/update theunimodular matrix T′ 120 so long as a diagonal element of the covariancematrix {tilde over (C)} are reducible. In an exemplary embodiment of thepresent invention, a diagonal element {tilde over (C)}_(k,k) of thecovariance matrix is reducible if an only if there exists an i≠k, suchthat λ_(i,k)≠0 i.e. R_(i,k)>0. In another exemplary embodiment of thepresent invention, a diagonal element {tilde over (C)}_(k,k) isreducible if and only if there exists a reduced value of {tilde over(C)}_(k,k), which is larger than a small value ε, as shown in Equation13.

R _(i,k)=−|λ_(i,k)|² {tilde over (C)} _(i,k)−λ*_(i,k) {tilde over (C)}_(k,i)>ε  Equation 13:

The present invention provides several strategies for selecting theindex pair (i, k) for each iteration, and thus, which element of thecovariance matrix will be reduced. In an exemplary embodiment of thepresent invention, reducing diagonal elements comprises reducing thelargest reducible diagonal element of the covariance matrix (FIG. 2provides exemplary pseudo code for this exemplary embodiment of thepresent invention). In another exemplary embodiment of the presentinvention, reducing diagonal elements comprises reducing the smallestreducible diagonal element of the covariance matrix. In yet anotherexemplary embodiment of the present invention, reducing diagonalelements comprises reducing a random reducible diagonal element of thecovariance matrix. In still yet another exemplary embodiment of thepresent invention, reducing diagonal elements of the covariance matrixcomprises reducing a reducible element of the covariance matrix thatrequires the least cost to be found. As those skilled in the art wouldunderstand, the “cost” can be determined many ways. In an exemplaryembodiment of the present invention, the “cost” can be the number ofarithmetic operations. In another exemplary embodiment of the presentinvention, the “cost” can be the number of clock cycles.

In addition to LR methods, various embodiments of the present inventionprovide LR systems. An exemplary embodiment of the present inventionprovides an LR system comprising a plurality of input terminals, aprocessor, and logic. The plurality of input terminals can be configuredto obtain an input linearly transformed by a transformation matrix. Thelogic can be stored in memory. The memory can be many types of memoryknown in the art. In an exemplary embodiment of the present invention,the logic is stored in a non-transitory computer readable media. Invarious embodiments of the present invention, the processor can beconfigured to execute the logic. When executed, the logic can beconfigured cause the system to perform one or more of the steps of theexemplary lattice reduction methods described herein.

FIGS. 3A-3C depict the BER performance of a ZFD and MMSE-D for singlecarrier frequency division multiple access (“SC-FDMA”) and LR-aided ZFDsfor generalized SC-FDMA (“GSC-FDMA”) with an exemplary LR system of thepresent invention, which is denoted by ELR-aided ZFD, GSC-FDMA, for4-(FIG. 3A), 16-(FIG. 3B), and 64-(FIG. 3C) order quadrature amplitudemodulation (“QAM”) schemes. The performance of an MLD for GSC-FDMA isalso plotted as a benchmark. For GSC-FDMA, the subcarriers for each userare divided into G=8 groups with each group having P=4 subcarriers. Asshown, ZFD for SC-FDMA achieves diversity 1. At 4-QAM, the MMSE-Dachieves higher order “diversity” from BER=10⁻² to 10⁻⁵ and obtainsclose performance to LR-aided ZFDs at BER=10-5. The MMSE-D, however,loses its advantage for 16-QAM and 64-QAM, where the diversity order 1is clear in these two cases. The LR-aided ZFDs and the exemplaryembodiment of the present invention have close performance to each otherand achieve significant improvement relative to ZFD and superiorperformance to MMSE-D for 16-QAM and 64-QAM. Compared with MLD, however,there still exists a 5 to 10 dB gap at BER 10⁻⁵.

FIG. 4 depicts the BER performance of a CLLL-aided MMSE-D, SA-aidedMMSE-D, and exemplary embodiments of the present invention, which aredenoted as ELR-aided MMSE-D, with different group sizes and 64-QAM. Whenthe group size is P=8, all the MMSE-Ds exhibit similar performance. Whenthe group size increases to P=32, the performance of the CLLL-aidedMMSE-D becomes worse and has more than a 5 dB the gap to SA-aided MMSE-Dat BER=10⁻⁴. The exemplary embodiment of the present invention,ELR-aided MMSE-D, however, exhibits substantially similar performance tothe SA-aided MMSE-D.

FIG. 5 depicts the BER performance of a ZFD and MMSE-D for SC-FDMA, aCLLL-aided ZFD and SA-aided ZFD for GSC-FDMA, and an exemplaryembodiment of the present invention, which is denoted by ELR-aided ZFD,for GSC-FDMA. In FIG. 5, L=8 channel taps, group size P=8, and 64-QAM.The exemplary embodiment of the present invention (ELR-aided ZFD)performs better than CLLL-aided ZFD and SA-aided ZFD, where theexemplary embodiment has about a 1.5 gain relative to CLLL-aided andSA-aided at BER=10⁻⁵. Further, as the SNR increases, the performance ofSA-aided ZFD degrades, and the gap to the exemplary embodiment of thepresent invention increases to about 10 dB at BER=10⁻⁶.

Various embodiments of the present invention also provide LR systems andmethods with lower complexity than conventional systems and methods.Table 1 (below) and FIG. 6 summarize the average number of basis updateiterations and number of arithmetic operations (real additions and realmultiplications) for basis updates, respectively, for a CLLL method, anSA method, and an exemplary embodiment of the present invention (denotedby ELR), in GSC-FDMA channel with L=4. The CLLL algorithm has thehighest number of basis updates of all three LR methods. Compared withCLLL, the exemplary embodiment of the present invention uses fewerupdates and SA exhibits the lowest bases updates among the three LRmethods. When it comes to the number of arithmetic operations, however,SA has a higher number of operations than either CLLL or ELR. This isbecause SA requires (128n−18) arithmetic operations per basis update,which is much higher than CLLL or ELR. The exemplary embodiment of thepresent invention (ELR) uses (31n−7) arithmetic operations per basisupdate, including (16n+8) to update λ_(i,k)and R_(i,k). The exemplaryembodiment of the present invention (ELR) achieves the lowest complexitywith respect to the number of arithmetic operations for basis updates.

TABLE 1 Group Size 4 8 16 32 CLLL 3.25 9.39 29.56 101.42 SA 2.09 3.796.56 15.02 ELR 2.37 5.17 10.92 24.12

FIGS. 7-15 illustrates performance comparisons between conventionalsystems and methods and exemplary embodiments of the present invention(denoted by D-ELR, D-ELR-aided MMSE-SIC, D-ELR-aided MMSE-SIC, andD-ELR-aided MMSE-SV-SIC) in various MIMO systems. FIG. 7 provides plotsof CCDF of od(H) and od({tilde over (H)}) in MIMO systems with N=M=4 forconventional methods and an exemplary embodiment of the presentinvention. FIG. 8 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with 4 QAM, SNR=20 dB, and different numbersof antennas. FIG. 9 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with 64 QAM, SNR=30 dB, and different numbersof antennas. FIG. 10 provides plots of the average number of basisupdates for conventional methods and an exemplary embodiment of thepresent invention in MIMO systems with different numbers of antennas.FIG. 11 provides plots indicating performance comparisons ofconventional methods and an exemplary embodiment of the presentinvention in MIMO systems with 256 QAM. FIG. 12 provides plotsindicating performance comparisons of conventional methods and anexemplary embodiment of the present invention in MIMO systems with 64QAM, SNR=30 dB, and different numbers of antennas. FIG. 13 providesplots indicating performance comparisons of conventional methods and anexemplary embodiment of the present invention in MIMO systems withM=N=64 and 256 QAM. FIG. 14 provides plots indicating performancecomparisons of conventional methods and an exemplary embodiment of thepresent invention in MIMO systems with M=N=32 and differentconstellation sizes. FIG. 15 provides plots indicating performancecomparisons of conventional methods and an exemplary embodiment of thepresent invention in MIMO systems with M=N=32, 64 QAM, and differentSNRs.

Table 2 (below) summarizes the average number of basis updates for anLLL method, a D-LLL method, an SA method, and an exemplary embodiment ofthe present invention (denoted by D-ELR), in for an exemplary MIMOsystem.

TABLE 2 N = M 4 8 16 32 64 128 LLL 6.70 29.19 97.24 248.91 561.941272.03 D-LLL 7.39 34.37 103.37 208.66 390.76 766.34 SA 5.50 16.98 33.8565.87 141.68 329.30 D-ELR 5.27 14.82 32.90 67.90 135.52 272.77

It is to be understood that the embodiments and claims disclosed hereinare not limited in their application to the details of construction andarrangement of the components set forth in the description andillustrated in the drawings. Rather, the description and the drawingsprovide examples of the embodiments envisioned. The embodiments andclaims disclosed herein are further capable of other embodiments and ofbeing practiced and carried out in various ways. Also, it is to beunderstood that the phraseology and terminology employed herein are forthe purposes of description and should not be regarded as limiting theclaims.

Accordingly, those skilled in the art will appreciate that theconception upon which the application and claims are based may bereadily utilized as a basis for the design of other structures, methods,and systems for carrying out the several purposes of the embodiments andclaims presented in this application. It is important, therefore, thatthe claims be regarded as including such equivalent constructions.

Furthermore, the purpose of the foregoing Abstract is to enable theUnited States Patent and Trademark Office and the public generally, andespecially including the practitioners in the art who are not familiarwith patent and legal terms or phraseology, to determine quickly from acursory inspection the nature and essence of the technical disclosure ofthe application. The Abstract is neither intended to define the claimsof the application, nor is it intended to be limiting to the scope ofthe claims in any way. It is intended that the application is defined bythe claims appended hereto.

1. A lattice reduction method comprising: obtaining a preliminaryestimate of a transformation matrix; generating a covariance matrixbased on the preliminary estimate of the transformation matrix; reducingdiagonal elements of the covariance matrix to generate a unimodulartransformation matrix; and using the unimodular transformation matrix toobtain an estimate of an input.
 2. The method of claim 1 furthercomprising repeating the step of reducing the diagonal elements untilthe diagonal elements of the covariance matrix are not reducible.
 3. Themethod of claim 1, wherein reducing diagonal elements of the covariancematrix comprises reducing a largest reducible diagonal element of thecovariance matrix.
 4. The method of claim 1, wherein reducing diagonalelements of the covariance matrix comprises reducing a smallestreducible diagonal element of the covariance matrix.
 5. The method ofclaim 1, wherein reducing diagonal elements of the covariance matrixcomprises reducing a random reducible diagonal element of the covariancematrix.
 6. The method of claim 1, wherein reducing diagonal elements ofthe covariance matrix comprises reducing a reducible diagonal element ofthe covariance matrix requiring the least costs to be found.
 7. Themethod of claim 1, wherein reducing diagonal elements of the covariancematrix comprises an iterative process.
 8. The method of claim 1, whereinthe transformation matrix comprises a channel matrix.
 9. The method ofclaim 1 further comprising obtaining a signal at a plurality of inputterminals, the signal representing an input linearly transformed by thetransformation matrix.
 10. The method of claim 1, wherein obtaining apreliminary estimate of the transformation matrix comprises obtainingdata from a memory.
 11. A lattice reduction system comprising: aplurality of input terminals configured to obtain a signal, the signalrepresenting an input linearly transformed by a transformation matrix; aprocessor; and logic stored in a non-transitory computer readable mediathat, when executed by the processor, is configured to: obtain apreliminary estimate of the transformation matrix; generate a covariancematrix based on the preliminary estimate of the transformation matrix;reduce diagonal elements of the covariance matrix to generate aunimodular transformation matrix; and use the unimodular transformationmatrix to obtain an estimate of the input.
 12. The system of claim 11,wherein the logic is further configured to repeat the step of reducingdiagonal elements of the covariance matrix until the diagonal elementsare not reducible.
 13. The system of claim 11, wherein the logic isfurther configured to reduce diagonal elements of the covariance matrixby reducing a largest reducible diagonal element of the covariancematrix.
 14. The system of claim 11, wherein the logic is furtherconfigured to reduce diagonal elements of the covariance matrix byreducing a smallest reducible diagonal element of the covariance matrix.15. The system of claim 11, wherein the logic is further configured toreduce diagonal elements of the covariance matrix by reducing a randomreducible diagonal element of the covariance matrix.
 16. The system ofclaim 11, wherein the logic is further configured to reduce diagonalelements of the covariance matrix by reducing a reducible diagonalelement of the covariance matrix requiring the least costs to be found.17. The system of claim 11, wherein the logic is further configured toreduce diagonal elements of the covariance matrix by iterativelyreducing at least one diagonal element of the covariance matrix.
 18. Thesystem of claim 11, wherein the transformation matrix comprises achannel matrix.
 19. A lattice reduction method comprising: obtaining apreliminary estimate of a transformation matrix; generating a covariancematrix based on the preliminary estimate of the transformation matrix;and reducing diagonal elements of the covariance matrix to generate aunimodular transformation matrix.
 20. The lattice reduction method ofclaim 19, wherein obtaining a preliminary estimate of a transformationmatrix comprises accessing data from a memory.